Geometry Problems of the Day (Geometry Regents, June 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2013

Part I: Each correct answer will receive 2 credits.


11. In ABC, D is the midpoint of AB and E is the midpoint of BC. If AC = 3x - 15 and DE = 6, what is the value of x?

1) 6
2) 7
3) 9
4) 12

Answer: 3) 9

If DE is the midsegment and DE = 6 then AC = 12 because the side of the triangle parallel to the midsegment is double the size of the midsegment. (And the midsegment is half the size of that side.)

So 3x - 15 = 12
and 3x = 27
then x = 9.

12. What are the coordinates of the center of a circle if the endpoints of its diameter are A(8,-4) and B(-3,2)

1) (2.5,1)
2) (2.5,-1)
3) (5.5,-3)
4) (5.5,3)

Answer: 2) (2.5,-1)

If you are given the endpoints of a diameter then the center of the circle is the midpoint. Average the two x-coordinates and the two y-coordinates.

Point M( (8 + -3)/2, (-4 + 2)/2 ) = (2.5, -1)

13. Which graph could be used to find the solution to the following system of equations?

y = (x + 3)² - 1
x + y = 2

Answer: 2) [SEE IMAGE]

The vertex of the parabola is (-3, -1). You don't have to worry about flipping the signs because all of the choices showing the vertex at x = -3. You DO NOT flip the sign of the y-coordinate. Subtracting 1 lowers the graph. Eliminate Choices (3) and (4).

The line x + y = 2 can be rewritten as y = -x + 2, which has a slope of -1, which goes DOWN as it moves to the right.

14. What is the converse of “If an angle measures 90 degrees, then it is a right angle”

1) If an angle is a right angle, then it measures 90 degrees.
2) An angle is a right angle if it measures 90 degrees.
3) If an angle is not a right angle, then it does not measure 90 degrees.
4) If an angle does not measure 90 degrees, then it is not a right angle.

Answer: 1) If an angle is a right angle, then it measures 90 degrees.

For Converse, think of sneakers, and walking around. The converse is when the IF walks over and becomes the THEN and vice versa. INVERSE is the one with the "Nots" (or the Nots removed).

Choice (1) has the two clause reversed. This is the answer.

Choice (2) is a closed statement of fact, which is true. But it is not a conditional statement, so it is not the Converse of the original statement.

Choice (3) is the Contrapositive. The clauses are switched but they are also negated.

Choice (4) is the Inverse. The clauses are negated, not switched.

15. As shown in the diagram below, a right pyramid has a square base, computations. ABCD, and EF is the slant height.

Which statement is not true?

1) EA ≅ EC
2) EB ≅ EF
3) triangle AEB ≅ triangle BEC
4) triangle CED is isosceles

Answer: 2) EB ≅ EF

It's a right pyramid with a square base.

Choice (1) is always true. These lines will always be congruent.

Choice (2) is never true. EF is the slant height and EB is an edge. EFB forms a right triangle and EB is the hypotenuse, which means it must be longer that EF.

Choice (3) is always true. The four triangular faces will always be congruent.

Choice (4) is always true. The four triangular faces will always be isosceles.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.